Abstract
Let E nbe the error norm of the best L ∞ rational approximation of degree n to the exponential function exp(-t) on [0,∞). Grounds are given for setting the conjectured limit E n / q n →2q 1/2 when n →∞, where q is the known constant ‘l/9’= 1/9.2890254919208189187554494359517450610316948677…, based on the singular values and functions of the relevant Henkel operator (Carathéodory-Fejér’s method).
Moreover, hints are given according to which a valuable asymptotic expansion of E„ should also contain n th powers of new constants q 1 = ‘1/56’, q 2 = ’ 1/ 240’, etc.
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Magnus, A.P. (1994). Asymptotics and Super Asymptotics for Best Rational Approximation Error Norms to the Exponential Function (The ‘1/9’ Problem) by the Carathéodory-Fejér Method. In: Cuyt, A. (eds) Nonlinear Numerical Methods and Rational Approximation II. Mathematics and Its Applications, vol 296. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0970-3_14
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